Hi, I'm an engineering student in my 2nd year, not that that's relevant but nevertheless, I stumbled across some interesting problems today that may be a little thought provoking.

The first is finding the first derivative of x^x
My initial guess is using the original basic differentiation technique that is

d (x^n) / dx = n x^(n-1)

and using the chain rule I believe (although it is inconesquential in this case since dx/dx = 1)
I came up with the answer to d x^x / dx as

x(x^(x-1)) ... which just happens to be x^x unless my brain has gone wonkey, regardless it looks very wrong to me, so perhaps someone can set me straight on that ^-^

The other thing was regarding this, i^i (i denotes the imaginary unit)

A friend told me today that this has infinite solutions, and that they are all real. Is anyone aware of any simple proof of this?
Regard, Yax